 Solution

The divergence and scalar vorticity areThus the flow is irrotational but not incompressible on the plane. A velocity potential satisfying $\mathbf{u}=\nabla \varphi$ isIndeed its two partial derivatives are the two velocity components. A usual two-dimensional stream function would satisfy $u_x={\psi }_y$, $u_y=-{\psi }_x$ and hence have identically zero divergence. No such stream function exists on an open region where this divergence is not identically zero; the line $y=0$ is not a two-dimensional flow domain.